Assignment 3: blood oxygen uptake. Lars Johansson IEI/mechanics Linköping university

Transcription

1 Assignment 3: blood oxygen uptake Lars Johansson IEI/mechanics Linköping university 1

2 Introduction In the human lung, the airway branches to the terminal structures which are the alveoli. The alveoli are in close contact with capillaries so that air is separated from blood by a very thin tissue membrane, see Figs. 1 and 2. Figure 1: Terminal part of the airways. Picture from Wikipedia. Figure 2: Alveoli-capillary gas exchange. Picture from Wikipedia. Inhumanblood,someoxygeniscarriedinplasmasolutionbutmostofthe oxygen is carried bound chemically to hemoglobin. The curve showing the saturation fraction of the oxygen binding sites of hemoglobin as a function of plasma oxygen concentration at steady-state is the well known oxygen dissociation curve. 2

3 Figure 3: The icefish Chionodraco hamatus. Picture from Wikipedia. Certain arctic fish, such as the Chionodraco hamatus, Fig. 3, have blood without hemoglobin or other oxygen carrier working by chemical binding. This means they have to rely solely on oxygen carried in solution in the blood plasma. In the first part of this assignment we will study what oxygen transport would be like with human physiological parameters, but without red blood cells, i.e. without hemoglobin. In the second part we will include transport of oxygen bound to hemoglobin. 3

4 Part one: the human icefish Figure 4: Idealized geometry. In this part of the assignment we will study what oxygen transport would be like with human physiological parameters, but without red blood cells, i.e. without hemoglobin. First, the geometry of Figs. 1 and 2 are idealized as shown in Fig. 4. The capillary is represented as a plane rectangle, i.e. the problem is solved in two space dimensions and the thickness in the z-direction is assumed to be one unit of length for the purpose of dimensional analysis. The oxygen concentration is governed by the diffusion-convection equation, while an alveolus is included as a boundary condition on oxygen flow into the capillary at a section of the capillary wall. The diffusion-convection equation is written: c t +v c = D 2 c (1) where D [m 2 /s] is the diffusion constant and c [mol/m 3 ] is the oxygen concentration to be calculated. Note that c is taken to be the total oxygen concentration of the flowing blood, but in this part of the assignment this is the oxygen concentration of plasma, since at this point we consider plasma without red blood cells. The plasma velocity v [m/s] brings oxygen along in convective transport, in addition to the diffusion. It is taken to be a known constant velocity in the x-direction: 4

5 [ u0 v = 0 The oxygen flow from the alveolus into the capillary is modelled as: ] (2) N n = P ac (p alv H c) (3) Here n is the outward normal of the capillary wall so that N n [mol/(m 2 s)] is the inward oxygen flow, p alv [Pa] is the partial pressure of oxygen in the alveolus, assumed to be constant, H [mol/(m 3 Pa)] is the solubility constant and P ac [m/s] is the permeability coefficient of the tissue membrane between the alveolus and capillary. Assignment A Implement a model as outlined above in COMSOL 3.x. The convergence of the numerical calculation will improve by using a reasonable guess of the steady-state distribution when setting initial conditions. One possibility is to to use the inlet (upstream) boundary condition as the initial value for the concentration in the entire domain; an improvement to this could be to use the alveolus concentration from the start of capillary-alveous contact and downstream. See Appendix A for further pointers on implementation, Appendix B for numerical data and Appendix C for instructions on how to submit the report. Submit plots of the concentration for flow velocities u 0 = [m/s] and u 0 = [m/s] at a time when steady-state conditions have developed. 5

6 Part two: the red-blooded human To add the transport of oxygen bound to hemoglobin to that dissolved in plasma, we replace Eq. 1 by the system: c t +v c = D 2 (c c 2 ) (4) c 2 t +v c 2 = 1 τ c 2 + k τ max(0, ( c c2 c 50 ) m 1+( c c 2 c 50 ) m) (5) to be solved simultaneously using the Multiphysics feature of COMSOL 3.x. Here, c [mol/m 3 ] is the total oxygen concentration in blood and c 2 [mol/m 3 ] is the part of the oxygen concentration of blood carried bound to hemoglobin, so that c c 2 [mol/m 3 ] is the part of the concentration carried in plasma solution. Further, τ [s] is a time constant to model the reaction kinetics when oxygen is bound to hemoglobin and m, c 50 [Pa] are constants in an approximation of the oxygen-hemoglobin dissociation curve. To understand the meaning of k [mol/m 3 ], consider a steady-state ( c 2 / t = 0) and homogenous ( c 2 = 0) condition so that the left hand side of Eq. 5 is zero: 0 = c 2 +kmax(0, ( c c 2 c 50 ) m 1+( c c 2 c 50 ) m) (6) The max operator is included to improve the numerics in the case c c 2 is negative, which is physically impossible but might occur numerically. Without the max operator, Eq. 6 gives c 2 = k (c c 2 c 50 ) m 1+( c c 2 c 50 ) m (7) Under steady state homogenous conditions, the hemoglobin carried oxygen concentration c 2 [mol/m 3 ] can be calculated from the plasma oxygen concentration c c 2 using Eq. 7; this should be exploited when setting the upstream boundary conditions and the initial conditions. It is also seen that for large values of plasma oxygen concentration c c 2, i.e. when the hemoglobin is 100% saturated, the hemoglobin carried oxygen c 2 [mol/m 3 ] has the value k. Inall, Eq. 5 will result in c 2 having a valueaccording toanapproximation of the dissociation curve if the calculation is allowed time to settle to a homogenous steady-state. In Eqs. 4 and 5 it is assumed that the red blood cells consists mainly of fluid with properties similar to plasma, and that the resistance to diffusion of the red blood cell walls can be neglected. Thus, blood is pictured as hemoglobin dissolved directly in plasma. 6

7 Assignment B Modify Eq. 3 to the form appropriate to use with Eqs. 4 and 5. Assignment C Discuss appropriate boundary and initial conditions. Assignment D Implement your modified model in COMSOL 3.x. Assignment E Submit plots of c 2 for flow velocities u 0 = [m/s], u 0 = [m/s] and u 0 = [m/s]. What is the saturation (fraction of hemoglobin carried oxygen concentration to the largest possible value according to Eq. 7) for these velocities? 7

8 Appendix A: pointers on implementation Pointers for part one of the assignment Start your COMSOL 3.x system. By default, the Model Navigator window is automatically launched; if not, select Multiphysics - Model Navigator (Fig. 6, arrow C) from the main interface. In the Model Navigator, select COMSOL Multiphysics - Convection and Diffusion - Convection and Diffusion - Transient Analysis, (Fig. 5, arrow C). The name of the dependent variable(that function of time and position which is to be calculated) can be changed (5D), but the default is OK; it is a virtue to keep all names short, since the boxes where things will be defined in the graphical user interface are quite small. We will make calculations in the plane, so keep the default Space dimension: 2D selection (5A). If you have saved previous work it can be opened here using Open (5B), or from the main interface. The Multiphysics (5F) feature can be used to define several partial differential equations on the same domain, but we defer this to the second part of the assignment when the Multiphysics feature will be invoked from the main interface. Next, use OK (5E) to launch the main interface. Figure 5: The Model Navigator window that opens when COMSOL 3.x is started. 8

9 In the main interface (Fig. 6), start by changing the scales to accomodate the geometry of our model. Select Options - Axes/Grid Settings (6A). Since our geometry is long and thin, deselect the default Axis equal setting (6B) to allow different scales in the x- and y-direction. When previous work is reopened from file, the Axis equal setting might have been automatically reset so that you have to deselect it again. Figure 6: The main interface. Use different scales on x- and y- axes. Next, we define the geometry of our model. Rectangles are drawn by selecting rectangle (7D) and drawing the rectangle by moving the mouse while holding down the left button. If the rectangle symbol (7D) is missing from the user interface, select the Draw Mode at (7C) or use Draw - Draw Objects - Rectangle/Square (7B). Do not draw the entire domain as one single rectangle. Instead, draw three separate but adjacent rectangles, the middle one being just as wide as the alveolus-capillary contact. Then unite the three rectangles by selecting Union (7E) after having selected all three rectangles by Edit - Select All or encircling them using the mouse pointer. All rectangles should be red directly after using Union and the three different rectangle names (R1, R2, R3 in Fig. 7) should be automatically replaced by a single name. The purpose of this indirect definition of the domain is to have a 9

10 separate part of the boundary where the alveolus-capillary oxygen diffusion can be specified. At this point you should start saving your work using File - Save As (7A). The native COMSOL format (.mph) is preferable for the present purposes since it is the fastest to load, much faster if you have done many changes to your work. Figure 7: Create a domain by uniting three rectangles. To define the partial differential equation to be solved, select Physics - Subdomain Settings (8C). A window will appear (8E) with the equation that was implicitly selected previously in the Model Selector window (5C). To model our particular problem, the equation is further specified by specifying equation parameters in the table (8G). Note that the equation must be specified separately for each subdomain (8D). The subdomains can be specified together by selecting Select by group. Also, specify initial conditions using Init (8F). Do not put numerical values in the Subdomain Settings window. Instead, invent names for all constants and define the constants in the Constants window, which you open using Options - Constants, (8A), (8B). Finally we need to specify the boundary conditions. Select Physics - Boundary Settings. Select type of boundary condition(9f) for each part of the boundary(9e). Do not put numerical values in the Boundary Settings window, put them in the Constants window. Equations can be used in the boundary condition specifications; this must be used for the boundary representing alveolar contact, where the flux depends on the plasma oxygen concentration 10

11 Figure 8: Defining the PDE to be solved. (Eq. 3 above). There might be a warning for inconsistent units here, since COMSOL does not recognize the units of constants you have defined. We are now ready to run a simulation. Select = (9D) or use Solve - Solve Problem (9B). To change parameters in the solver, such as the time interval, use Solve - Solver Parameters (9B). The default simulation time is one second; it is likely that you will have to increase this for some of your calculations. Also, if it is difficult to get the numerical calculation to converge, and any possibility of an implementation error has been ruled out, you might want to tighten the tolerances of the solver. If the solution is sucessefull, a colour plot of the solution at the end of the solution interval will appear. You can find numerical values by pointing with the mouse and pushing the left button. If you want to modify the plot, for example viewing some other quantity than default or at a different time, use Postprocessing - Plot Parameters (9C). 11

12 Figure 9: Specify boundary conditions and run a simulation. Pointers for part two of the assignment In the second part of the assignment, two partial differential equations will be solved simultaneously on the same domain. To this end, with your model from the first part loaded in the main interface, start the Model Navigator using Multiphysics - Model Navigator (10C). Select COMSOL Multiphysics - Convection and Diffusion - Transient Analysis (10A), which should be the default. You have now selected a new PDE, which is of the same type as the one already in use, but independent from it. Add the new equation using Add (10D). The unknown function added to the problem by adding another PDE will be called c2 if you keep the default name (10B). Leave the Model Navigator using OK. To specify the new PDE and modify the old one, Physics - Subdomain Settings and Physics - Boundary Settings are again used. Before opening one of these settings windows, you must be careful to select in the Multiphysics menu (10C) which one of the two equations you wish to modify. The two PDE s now defined can be coupled so that the unknown function c2 of the 12

13 Figure 10: Add a second PDE to be solved simultaneously with the first. second PDF can appear in the definition of the first and vice versa. The nonstandard terms on the right hand sides of our equations are implemented using the source term R. Be careful to specify priority by using enough parentheses, particularly the exponential operator is a bit dangerous. Partial derivatives are denoted by trailing x or y to the function name as appropriate so that, for example, applying the Laplace operator to c2 is expressed as 2 c2/ x c2/ y 2 = c2xx + c2yy. 13

14 Appendix B: numerical data These are quantities that are difficult to measure. The numerical values given are approximate at best and guesses at worst. l ac = [m] h = [m] P ac = P ac [m/s]. Here P ac is your date of birth divided by 10000; if, for example, you were born on the 26:th of the month, your permeability is P ac = [m/s] H = 10 5 [mol/(m 3 Pa)] D = [m 2 /s] p alv =? [Pa] Find a value of the oxygen partial pressure in the alveoli appropriate for a healthy human breathing athmospheric air and give a reference in your report. Wikipedia or an.html document is not an acceptable reference. p v =? [Pa] Find a value of the oxygen tension in the blood just before entering the lungs appropriate for a healthy human breathing athmospheric air and give a reference in your report. Wikipedia or an.html document is not an acceptable reference. k = 9.2 [mol/m 3 ] m = 3 [dimensionless] c 50 = [mol/m 3 ] τ = [s] 14

15 Appendix C: instructions for submitting the report For the 2014 course, the assignment report is submitted as a pdf file to jonst11.liu@analys.urkund.se. Please name the file with the asignment number and your name, for example: assignment3-your-name.pdf. The report can be a maximum of seven pages long (with minimum font size 11 pt and normal text margins) or, alternatively, a maximum of four pages plus five additional plots in an appendix. 15